

Tweet
Follow @sheldonbrowncom 
Conventional bicycle wheels are spoked using a semitangent, crossed spoking pattern. Each spoke crosses one or more other spokes on the same side of the wheel, and there are leading and trailing spokes at the hub. To transmit power, the leading spokes' tension decreases and the trailing spokes' tension increases; vice versa when a hub brake is applied.
If a wheel has a large hub, though, crossed spokes may approach the rim at a sharp angle which the spoke holes and spoke nipples cannot accommodate. This is a common problem with electricbike hub motors.
On the other hand, with radial (cross zero) spoking, (spokes pointing directly outward from the hub), a wheel which transmits torque "winds up," as described in the main wheelbuilding article. Excess motion leads to spoke failure, and to the risk of pulling apart hub flanges due to excessive spoke tension. Wheels with large hubs are sometimes laced radially with extrathick spokes, but still, spoke breakage has occurred.
Within limits, the problem of excessive spoke angles can be solved by reducing the cross number of a conventional spoking pattern. The wheel shown below is pushing the limit with a crossone pattern, the most forgiving conventional pattern which can transmit torque . Even in this fullytensioned wheel, the spokes are bending as they leave the spoke nipples.
Another approach has been to place leading spokes on one side of the wheel, and trailing spokes on the other, so the spokes can be more nearly radial. In this spoking pattern, none of the spokes on either side of the rim cross over each other, but spokes on opposite sides of the wheel do.
(Image from Golden Motor: fair use as commentary under U.S. copyright law)
This pattern converts the windup problem into other problems:
What is to do?
An answer is found in any of several unconventional spoking patterns. All have cross numbers less than cross one but more than cross zero (radial): the angle of the spokes as they approach the rim is a fraction of what it would be for cross one. Cyclist and engineer Damon Rinard has described on this site how to calculate spoke lengths for fractional cross numbers, and provided a spokelength calculator which allows these combinations. Several spoke patterns are described below, suitable for different rims and hubs.
Warning, though: All of these patterns use nearradial spokes, and some use hubs with high spoke counts. These patterns should only be used with large hubs with strong flanges. There is no reason anyway to use them with smaller hubs, and also, the resistance to torque may be too low with these patterns and small hubs.
Some of these patterns have all radial spokes on one side. These patterns should not be used when torque is applied at both sides (for example, with a freewheel on one side and a disc brake on the other) unless the hub barrel is stout and strong.
Also: the math for some of the patterns needs checking and testing by building wheels. Please recalculate before using the information here. sheldonbrown.com cannot be held responsible for incorrect spoke lengths.
The patterns are listed in the table below, with links to full descriptions later in this article. I suggest that you read through the first section below the table before selecting a pattern, to learn how these patterns work.
Type of pattern  Spoke holes in hub  Spoke holes in rim 

5:3 ratio  40  24 
4:3 ratio  32  24 
48  36  
3:2 ratio  36  24 
48  32  
5:4 ratio  40  32 
9:7 ratio  36  28 
2:1 ratio  48  24 
Four more holes in hub than in rim 
28  24 
32  28  
36  32  
40  36  
Spokes paired at hub  See description  Any mult. of 4 
Dished, 2 right:1 left  24 drilled to 48  36 
36 drilled to 72  36  
32 drilled to 64  48  
48 drilled to 96  48 
Mismatched spoke counts are practical when the number of spokes in the hub and rim are in small ratios of whole numbers, so the spokes form a repeating pattern. In the patterns described in this article, the hub has more spoke holes than the rim. All of the spoke holes in the rim are used, to build a strong wheel. These patterns require no modification to the hub or rim. They all use more than one length of spokes.
With most of these patterns, if the rim is of opposite handedness to what is shown, the valve hole should go into the next gap, left or right, between groups of spokes.
A 40hole hub and 24hole rim offer a good example. In the drawing below, we are looking at the right side of the wheel. Spokes and spoke holes shown in white are on the right side of the wheel, and those in dashed blue, on the left side. The valve is shown in green. The left and rightside spokes may be switched if the handedness (see explanation here) of the spoke holes in the rim would place the key spoke  just to the right of the valve in the image  on the left side at the rim.
With 5 spoke holes in the hub for every 3 in the rim, groups of 3 spokes at the hub are separated by two empty holes, one in each flange. Two holes at a time must be skipped so the next occupied spoke hole is on the opposite side from the previous one, as shown in the drawing.
The middle spoke of each group of 3 is radial, and the adjacent spokes are nearradial  not quite radial, thanks to the smaller angle between spoke holes in the hub, than in the rim.
With this pattern, two of every three spokes  all the nonradial ones  transmit torque. A leading spoke on the right side is adjacent to a trailing spoke on the left side, and this combination pulls the rim to the left at that pair of spokes. The next pair of spokes which transmits torque pulls the rim to the right. Eight pairs of spokes in all transmit torque, and so the rim tends to develop four "waves", compared with the six it would have with conventional spoking. Somewhat greater rim stiffness is needed, then, to avoid the rim's zigzagging from side to side under a torque load.
The fractional cross numbers for the nearradial spokes can be entered into Rinard's Spocalc spokelength calculator. The resulting spoke length will be the same, whether calculated based on the 24spoke rim or the 40spoke hub.
The cross numbers are:
40/24  Radial 
Nearradial 
Count  8  16 
Angle  0.000°  6.000° 
X, 24  0.000  0.200 
X, 40  0.000  0.333 
How are these numbers calculated?You may read the text in this box if the math interests you. You don't have to read it to make use of this article. In our 24hole rim, the angle between spoke holes is 360/24, or 15 degrees. The angle between spoke holes in the 40hole hub is 360/40 or 9 degrees. The outer spokes of our 3spoke groups are, then, 9 degrees away from the central, radial spoke at the hub, and 15 degrees away at the rim. The difference is 6 degrees. Cross 1 takes a spoke past one spoke hole in the rim to the second spoke hole, and so the angle to the second spoke hole is 30 degrees at the rim and 18 degrees at the hub. 6 degrees divided by the 30 degrees for cross 1 is 0.2, and so this is the cross number for the nearradial spokes, based on the 24spoke rim. 6 degrees divided by 18 degrees is 0.333 and so this is the cross number for the nearradial spokes, based on the 40spoke hub. You can compare the spoke lengths calculated both ways. They'll match if you have entered the right numbers into the spokelength calculator. 
Next, let's look at a wheel which has 4 spokes in the hub for every 3 in the rim. We'll work with a 32hole hub and a 24hole rim.
Because we must skip two spoke holes in a row in the hub, the spokes are in four groups of 5, 7, 5 and 7, as shown in the image below. The central spoke in each group is radial, and there are three different lengths of nearradial spokes. The odd number of spokes in each group cancels torque. If the spokes were in four groups of six, one flange would be tending to rotate the hub shell forward, and the other, backward, twisting the hub shell. Also torque from pedaling would pull the rim sideways, as described earlier.
Potential zigzagging of the rim is a bit irregular because of the different numbers of spokes in the groups, but there are four major waves. The greatest excursions are between the pairs of spokes which are farthest apart.
More mathAngles are calculated as follows: Spoke holes are 15 degrees apart at the 24hole rim, and 11.25 degrees apart at the 32hole hub. The angle to the second spoke hole for cross 1 is 30 degrees at the rim and 22.5 degrees at the hub. The difference is 3.75 degrees, and this is the angle for the spokes either side of the radial spoke. The next spoke is at twice that, 7.5 degrees, and the third spoke, three times, 11.25 degrees. 
The angles and cross numbers for the three different spoke lengths are given in the table below.
32/24  dist. from midline of 5 or 7spoke pattern  
radial  closest  middle  farthest  
Count  4  8  8  4 
Angle  0.000°  3.75°  7.5°  11.25° 
X, 24  0.000  0.125  0.250  0.375 
X, 32  0.000  0.167  0.333  0.500 
48/36 also uses 4 spoke holes at the hub for every 3 at the rim.
48/36  dist. from midline of 5 or 7spoke pattern  
radial  closest  middle  farthest  
Count  6  12  12  6 
Angle  0.000°  2.5°  5°  7.5° 
X, 36  0.000  0.125  0.250  0.375 
X, 48  0.000  0.167  0.333  0.500 
With the 36 spokes, there are six waves. This pattern works with a conventional 48spoke hub where the spoke holes are offset between the two flanges  not with a 24hole hub drilled with 24 additional holes. We have a pattern for that hub too, later in this article.
Combinations with 3 spokes in the hub to every 2 in the rim are also possible. Because we must skip two spoke holes at a time, we need alternating groups of 3 and 5 spokes to get 8 spokes in the rim for every 12 spoke holes in the hub. The uneven number of spokes in each group cancels torque on the hub shell, as described earlier. The drawing below shows the spoking pattern for a 36hole hub and 24hole rim.
The angles and cross numbers, calculated as in the previous examples, are
36/24  dist. from midline of pattern  
radial  middle  farthest  
Count  6  12  6 
Angle  0.00°  5.00°  10.00° 
X, 24  0.000  0.167  0.333 
X, 36  0.000  0.250  0.500 
A 48hole (conventional, not redrilled 24hole) hub and 32hole rim also can use this pattern.
48/32  dist. from midline of pattern  
radial  middle  farthest  
Count  8  16  8 
Angle  0.00°  3.75°  7.50° 
X, 32  0.000  0.167  0.333 
X, 48  0.000  0.250  0.500 
3 out of 4 spokes transmit torque, but those at the ends of the 5spoke groups are tensioned or slackened more when transmitting torque. This difference is reduced somewhat by the flexibility of the rim, and these spokes are on the right side, where the spokes are under higher tension in a dished wheel and so less likely to go slack due to torque.
This laces a 32hole rim to a 40hole hub, as shown in the illustration below.
The spokes are in groups of 7 and 9, with a radial spoke at the center of each as has already been described above for the 4/3 ratio. Five different lengths of spokes are required:
40/32  dist. from midline of 7 or 9spoke pattern  
radial  closest  second  third  fourth  
Count  4  8  8  8  4 
Angle  0.000°  1.125°  3.375°  5.625°  7.875° 
X, 32  0.000  0.05  0.15  0.25  0.35 
X, 40  0.000  0.062  0.187  0.313  0.438 
As with the 32/24 pattern, there are four waves.
A 9/7 ratio laces a 28spoke rim to a 36spoke hub, as shown in the illustration below.
Each quadrant of the wheel has 7 spokes with a central, radial spoke. Four different spoke lengths are required. Angles and cross numbers are as in the table below.
36/28  dist. from midline of 7spoke pattern  
radial  closest  middle  farthest  
Count  4  8  8  8 
Angle  0.00°  2.86°  5.71°  8.57° 
X, 28  0.000  0.111  0.222  0.333 
X, 36  0.000  0.143  0.286  0.429 
With torque from pedaling or a hub brake, the two topmost spokes and the other three pairs which are spaced most widely pull the rim sideways the most, and in alternate directions.
A 2/1 ratio results in a pattern as shown in the image below. This is for a 48spoke hub and 24spoke rim. All of the leftside spokes are radial. If the rim is of the opposite handedness, the valve moves clockwise by 3 spokes, rather than reversing the sides for the spokes  to keep the radial spokes on the left side.
Angles are as in the table below:
48/24  Left  Right 
Count  12  12 
Angle  0°  7.5° 
X, 24  0.0  0.250 
X, 48  0.0  0.500 
Rightside spokes alternate between leading and trailing, and so the rim tends toward 6 waves, same as with conventional spoking. Tension changes on the leading and trailing spokes due to torque are like those on any halfradial wheel.
36hole hubs and 32hole rims are common. The solution shown below uses two groups of spokes, one with 17 spokes, the other with 15. This wheel looks odd because the two groups have different numbers of spokes, but as with some of the patterns already described, groups of different sizes are necessary to avoid twisting the hub barrel. Each group is symmetrical around a radial spoke at its center. The spoke pattern results in a very slight wheel imbalance, but less than is probably produced by the valve.
The illustration below is for a wheel with a 36hole hub and 32hole rim. This pattern theoretically requires nine different spoke lengths, but the length of the radial spokes and adjacent ones may be close enough that the same length may be used.
The same principle may be applied to other combinations where the hub has 4 more spoke holes than the rim. All common spoke counts are covered in the tables below.
28/24  Radial  Next  Last  
Count  2  4  4  4  4  4  2 
Angle  0°  2.14°  4.29°  6.43°  8.57°  10.71°  12.86° 
X, 24  0.0  0.071  0.143  0.214  0.286  0.357  0.429 
X, 28  0.0  0.083  0.166  0.250  0.333  0.417  0.500 
32/28  Radial  Next  Last  
Count  2  4  4  4  4  4  4  2 
Angle  0°  1.61°  3.21°  4.82°  6.43°  8.04°  9.64  11.25° 
X, 28  0.0  0.063  0.125  0.179  0.250  0.313  0.375  0.438 
X, 32  0.0  0.071  0.143  0.214  0.286  0.357  0.429  0.500 
36/32  Radial  Next  Last  
Count  2  4  4  4  4  4  4  4  2 
Angle  0°  1.25°  2.5°  3.75°  5°  6.25°  7.5°  8.75  10.0° 
X, 32  0.0  0.056  0.111  0.167  0.222  0.278  0.333  0.389  0.444 
X, 36  0.0  0.063  0.125  0.179  0.250  0.313  0.375  0.438  0.500 
40/36  Radial  Next  Last  
Count  2  4  4  4  4  4  4  4  4  2 
Angle  0°  1.00°  2.00°  3.00°  4.00°  5.00°  6.00°  7.00°  8.00°  9.00° 
X, 36  0.0  0.050  0.100  0.150  0.200  0.250  0.300  0.350  0.400  0.450 
X, 40  0.0  0.056  0.111  0.167  0.222  0.278  0.333  0.389  0.444  0.500 
People who frequent this Web site know that we are not great fans of "boutique" wheels with paired spokes at the rim. Pairing spokes at the hub, however, offers an elegant way to avoid excessive spoke angles with a large hub and conventional rim.
The spoking pattern is essentially a radial pattern, except that uneven spacing of the spoke holes at the hub produces leading and trailing spokes. This pattern can produce any desired cross number by adjustment of the location of spoke holes in the hub.
This pattern is especially useful because it starts out with a hub and rim which have the same number of spoke holes. This can be any multiple of 4, as it is with all common hubs and rims.
Let's start our exploration of this pattern by first considering a 36spoke wheel, spoked radially. The spokes all point directly outward from the hub, as shown in the image below.
The wheel in the photo has radial spoking on a small hub. Our analysis for large hubs works best if the valve hole goes at zero degrees, halfway between the spokes at 355 and 5 degrees as shown in the table below.
Right flange  355  15  35  55  75  etc.  
Spoke            
Rim  345  355  5  15  25  35  45  55  65  75  85  etc.  
Spoke              
Left flange  345  5  25  45  65  85  etc. 
Let's now drill new spoke holes in the hub, halfway in between the predrilled ones. This is only safe  and only of any advantage anyway  if the hub is very large so the spoke holes are far apart. The hub flanges must be meaty enough to withstand a nearradial pull. There will now be 36 holes in each flange, including holes directly in line with those in the opposite flange.
Now, we rotate the hub by 5 degrees, so the angles of the spoke holes in the hub are between those in the rim, and we spoke the wheel as shown in the table and drawing below. The two spokes nearest the top are now 20 degrees apart at the hub, at 350 and 10 degrees, but they are still 10 degrees apart at the rim. We pair the remaining spoke holes in the hub as shown in the table below.
Left flange  350  20  30  60  70  etc.  
Spoke  \  /  \  /  \  
Rim  345  355  5  15  25  35  45  55  65  75  85  etc.  
Spoke  /  \  /  \  /  \  
Right flange  340  10  20  50  60  90  etc. 
The result is the pattern shown below.
36/72  Nearradial 
Offset  10° 
Count  36 
Angle  5° 
X, 36  0.250 
If you drill the hub with new spoke holes halfway between the all of the original ones, rim handedness is never a problem: use whatever spoke holes give the correct handedness. If the spoke holes in the rim are close to the centerline, matching handedness probably doesn't matter anyway, because a nearradial pattern is forgiving of misalignment of the spoke nipples.
As noted already, this pattern will work with a conventional rim, as the pattern repeats every four spokes. In fact, the spoke alignment is the same as with the outer ends of spokes in a conventional crossed pattern. Here's the same pattern for a 24hole rim, and showing only the holes which are used.
Left flange  345  30  45  90  105  etc.  
Spoke  \  /  \  /  \  
Rim  337.5  352.5  7.5  22.5  37.5  52.5  67.5  82.5  97.5  112.5  127.5  etc.  
Spoke  /  \  /  \  /  \  
Right flange  330  15  30  75  90  135  etc. 
24/48  Nearradial 
Offset  15° 
Count  24 
Angle  7.5° 
X, 24  0.250 
If the hub is predrilled with paired spoke holes, don't assume automatically that they will work with this pattern. The pairs on one flange need to be offset from those on the other, not directly opposite and not halfway in between the pairs on the other flange. The counterclockwise hole of a pair on one flange must be halfway between the clockwise holes of the adjacent pairs in the other flange (and vice versa). (Note how this is the case in the 24/48 drawing above. In each group of four spokes, the rightmost spoke goes to the left flange, and the leftmost one, to the right flange. The holes in the hub flanges for these spokes are evenly spaced.)
The photo below shows a 36spoke wheel with paired spoke holes of the hub close together, producing a cross number greater than 0.25. Though the holes nearer the middle of the group are not directly opposite each other as in our earlier examples, the hub meets our requirement for the outermost spoke holes in each group, The handedness of this wheel places the pairs of leftside spokes counterclockwise of the rightside ones, because the first spoke hole clockwise of the valve hole is toward the left side of the rim. The valve is visible at the lower right in the photo.
Image from electricbicyclesmagazine.com
Once having determined that the positions of spoke holes in a predrilled hub is correct, you need to measure and calculate the cross number depending on the spacing between paired spokes in the hub:Measure the angle or distance between the two spoke holes of a pair in one hub flange (holes which are closer together). We'll call that A.
Also measure the angle (or distance, that's accurate enough) between two holes which are farther apart. We'll call that B.
The cross number x is as
x = (B  2A)/2B
So, we see, for example, for a purely radial 36spoke wheel, A is 20 degrees and B is 40 degrees, so the cross number is
x = [40  (2 * 20)]/(2 * 40) = 0
For our 36hole wheel where we drilled additional spoke holes, so the holes are half as far apart as in a radial wheel, A is 10 degrees, so the cross number is
x = [40  (2 * 10)]/(2 * 40) = 0.25
If the paired spoke holes in the hub are brought closer and closer together, they approach being directly on top of one another. Then A would be 0 degrees, and the cross number would be
x = [40  (2 * 0)]/(2 * 40) = 0.5
And as a check on the math, for a normal cross 1 pattern, where A is 20 degrees,
x = [40  (2 * 20])/(2 * 40) = 1
Most wheels with a sprocket cluster must be dished to center the rim. Dishing reduces the tension of the spokes on the left side of a conventional wheel which has the same number of spokes on both sides. If the left hub flange is about twice as far from the centerline as the right flange, as is common, using one spoke on the left for every two on the right will nearly equalize the tension. The rim must have a number of spoke holes divisible by 3. The practical, available choice is a 36hole or 48hole rim. A 24hole rim would have only 8 spokes on the left, not enough with most rims.
The 36hole rim works with a 36hole hub drilled to have 72 spoke holes, 36 in each flange. There are then holes directly in line with one another in the two flanges. 24 spokes go to the the right flange of the hub, using paired spoking with spokes in 2 adjacent holes of every 3 in the right hub flange. The 12 spokes from the left flange alternate leading and trailing to avoid twisting the hub barrel. The difference in angle at the hub and rim for the spokes on both sides is 5 degrees, (compared with the 20 degrees of cross 1) so the spoke length is for a 0.25 cross, 36spoke wheel. This pattern is shown in the image below. The spokes are in repeating groups of 6, so this pattern only works for hubs and rims with a multiple of 6 spokes. Rim handedness doesn't matter, because it will always be wrong for one of the two spokes closest to the valve.
36/72  Nearradial 
Count  36 
Angle  5° 
X, 36  0.250 
A 48hole rim may also be used, similarly.
48/96  Nearradial 
Count  48 
Angle  3.75° 
X, 48  0.250 
The 2/1 pattern also works with a hub that starts with 2/3 as many spoke holes as the rim. You need to drill new spoke holes halfway between the original ones on both sides of the hub, and alternate leading and trailing spokes on the left side to avoid twisting the hub shell.. The image below shows a 36spoke wheel with a 24spoke hub modified by doubling the number of spoke holes. Rim handedness will always be wrong for some spokes.
36/48  Left  Right 
Count  12  24 
Angle  7.5°  2.5° 
X, 36  0.375  0.125 
X, 24  0.250  0.083 
A 48hole rim and 32hole hub may also be used, similarly.
48/64  Left  Right 
Count  16  32 
Angle  5°  1.67° 
X, 48  0.375  0.125 
X, 32  0.250  0.083 
36/48  Left 
Right 
Count  12  24 
Angle.  7.5°  12.5° 
X, 36  0.375  0.625 
X, 24  0.250  0.417 
This pattern also may be used with a 48spoke rim and a modified 32spoke hub:
48/64  Left 
Right 
Count  16  32 
Angle.  5°  8.33° 
X, 48  0.375  0.625 
X, 32  0.250  0.417 
All of the 2 right/1 left patterns send some spokes to the "wrong" side of the rim, and so they work better if the left and right spoke holes in the rim are near the centerline.
A wheel with a large hub, especially with a small rim, may need spokes shorter than any which are available at most bicycle shops.
There are three main ways to solve this problem.
Last Updated: by John Allen